Exact solution of the elliptic and hyperbolic near-neighbors Jastrow models beyond the ground state: excited eigenfunctions, spin generalizations, and correlation functions

The classification of Baradaran, Carrasco, Finkel and Gonzalez-Lopez (2018) determined all one-dimensional quantum potentials with nearest- and next-to-nearest-neighbors interactions whose ground state is Jastrow-like, i.e. a product of pair factors over consecutive particle separations; it proved that a three-body term is unavoidable and exhibited, as the principal new example, an elliptic near-neighbors potential built from Weierstrass zeta and wp functions on a circle, together with a new hyperbolic potential. For the previously known rational and trigonometric near-neighbors models the exact theory is essentially complete: closed-form excited eigenfunctions, spin generalizations yielding quasi-exactly solvable short-range spin chains, and the spacing statistics of the ground-state density (the intermediate spectral statistics of Bogomolny, Gerland and Schmit). For the new elliptic and hyperbolic families only the ground state and its energy are known. The problem posed here is to determine the exact solution content of the elliptic near-neighbors Hamiltonian and its hyperbolic degenerations beyond the ground state: (i) whether further eigenfunctions can be computed in closed form, explicitly or through quasi-exact solvability; (ii) the construction of spin versions and the associated short-range spin chains; and (iii) the correlation functions of the ground-state probability densities, including the elliptic deformation of the intermediate statistics.

Background

Exactly solvable one-dimensional quantum many-body systems of Calogero-Sutherland type are characterized by ground states of Jastrow form: a Jastrow wave function is a product of pair factors over all pairs of particles, $\psi(x_1,\dots,x_N)\propto\prod_{i<j}f(x_i-x_j)$, the rational (Calogero) and trigonometric (Sutherland) long-range models with inverse-square and $\sin^{-2}$ interactions being the canonical examples. A natural short-range counterpart restricts each particle to interact only with its nearest and next-to-nearest neighbors. In such near-neighbors interaction (NNI) models the Hamiltonian is

$$H=-\sum_i\frac{\partial^2}{\partial x_i^2}+\sum_i V_2(x_i-x_{i+1})+\sum_i V_3(x_i-x_{i+1},\,x_{i+1}-x_{i+2}),$$

with cyclic indexing on a circle of circumference $l$ (where $x_{N+1}=x_1-l$) or open indexing on the real line, and the ground state is Jastrow-like: $\psi\propto\prod_i\chi(x_i-x_{i+1})$, a product over consecutive pairs only, in contrast with the full product over all differences $x_i-x_j$ (the $A_{N-1}$ root system, i.e. the root system of the Lie algebra $A_{N-1}$ whose roots are the differences $e_i-e_j$) in the long-range case.

The rational and trigonometric NNI potentials were introduced by Jain and Khare (1999). Baradaran, Carrasco, Finkel and Gonzalez-Lopez (2018) then completely solved the corresponding classification problem, the near-neighbors analogue of the long-range classification of Koprucki and Wagner (2000), proving that every translation-invariant NNI potential admitting a Jastrow-like ground state must contain a genuine three-body term, and that the general solution depends on an essentially arbitrary odd function $\varphi$ with a simple pole at the origin (plus a confining parameter $\omega\ge 0$ on the line):

$$V(\mathbf{x})=\omega^2 r^2+2\sum_i\big(\varphi'(x_i-x_{i+1})+\varphi^2(x_i-x_{i+1})\big)-2\omega\sum_i(x_i-x_{i+1})\varphi(x_i-x_{i+1})-2\sum_i\varphi(x_i-x_{i+1})\varphi(x_{i+1}-x_{i+2}),$$

with ground state $\psi\propto e^{-\omega r^2/2}\prod_i\chi(x_i-x_{i+1})$, where $\chi(x)=\exp\int^x\varphi(s)\,ds$ and $r^2=\sum_i x_i^2$. The previously known rational ($\varphi(x)=\alpha/x$) and trigonometric ($\varphi(x)=(\pi\alpha/l)\cot(\pi x/l)$) models appear as particular cases, together with a new hyperbolic model on the line ($\varphi(x)=\alpha\beta\coth(\beta x)$ with $\omega>0$, the near-neighbors analogue of the long-range hyperbolic model of Forrester (1994) whose ground state is a Wigner crystal, i.e. a configuration in which the particles localize at the vertices of a regular lattice) and, as the main novelty, an elliptic model on the circle built from Weierstrass elliptic functions. Here $\varphi(x)=\alpha(\zeta(x)-2\eta_1 x/l)$ with $\alpha>1/2$, where $\zeta$ is the Weierstrass zeta function (the meromorphic, quasiperiodic function with $\zeta'=-\wp$), $\wp$ is the Weierstrass elliptic function with real period $l$ and imaginary half-period $\omega_3$ (so that $\operatorname{Im}\omega_3>0$), $\eta_1=\zeta(l/2)$, and $\sigma$ denotes the Weierstrass sigma function defined by $\sigma'/\sigma=\zeta$. The elliptic Hamiltonian (up to an additive constant) is

$$V=-2\alpha\sum_i\wp(x_i-x_{i+1})+2\sum_i\varphi(x_i-x_{i+1})^2-2\sum_i\varphi(x_i-x_{i+1})\varphi(x_{i+1}-x_{i+2}),$$

with Jastrow-like ground state

$$\psi\propto\exp\Big(-\frac{\alpha\eta_1}{l}\sum_i(x_i-x_{i+1})^2\Big)\prod_i|\sigma(x_i-x_{i+1})|^\alpha,\qquad E=\frac{4N\alpha\eta_1}{l}.$$

It degenerates to the trigonometric Jain-Khare model as $\operatorname{Im}\omega_3\to\infty$ and to a hyperbolic ($\coth$-type) potential in the opposite degeneration, so the elliptic model is the common source of all previously known NNI families.

For the rational and trigonometric predecessors the theory is essentially complete. Enciso, Finkel, Gonzalez-Lopez and Rodriguez constructed their spin versions through the exchange operator formalism (an algebraic framework in which operators permuting particle labels replace the momenta) and computed infinite families of excited eigenfunctions in closed form in terms of Laguerre and Jacobi polynomials (2005, 2007); via the freezing trick (the procedure that extracts a spin chain from a supersymmetric many-body model by letting the coupling grow so that the particles freeze at their classical equilibrium positions) they derived a quasi-exactly solvable nearest-neighbors spin chain intermediate between the Heisenberg chain and the Haldane-Shastry chain (the exactly solvable long-range spin chain with inverse-square interactions between all pairs of spins) (2008). A model is called quasi-exactly solvable if a finite-dimensional space of eigenfunctions can be computed in closed form, even though the rest of the spectrum cannot. On the statistical side, the probability density of the rational ground state, $\prod_i|x_i-x_{i+1}|^\beta$ (a short-range Dyson gas on the circle, i.e. the Coulomb-like statistical model for random-matrix eigenvalues with the pairwise interaction truncated to consecutive neighbors), was analyzed by Bogomolny, Gerland and Schmit (1999): its spacing distribution defines an intermediate spectral statistics between Poisson and Wigner-Dyson that describes pseudo-integrable billiards and the Anderson model at the metal-insulator transition.

For the new elliptic and hyperbolic families none of this exists: only the ground state and its energy are known.

Problem Statement

For the elliptic near-neighbors Hamiltonian $H=-\sum_i\frac{\partial^2}{\partial x_i^2}+V(\mathbf{x})$ on the circle of circumference $l>0$, with cyclic convention $x_{N+1}=x_1-l$ and

$$V(\mathbf{x})=-2\alpha\sum_i\wp(x_i-x_{i+1})+2\sum_i\varphi(x_i-x_{i+1})^2-2\sum_i\varphi(x_i-x_{i+1})\varphi(x_{i+1}-x_{i+2}),$$

where $\varphi(x)=\alpha\big(\zeta(x)-2\eta_1 x/l\big)$ with $\alpha>1/2$ and $\eta_1=\zeta(l/2)$, and $\zeta$, $\wp$ and $\sigma$ denote the Weierstrass zeta, elliptic and sigma functions (related by $\sigma'/\sigma=\zeta$ and $\zeta'=-\wp$) with real period $l$ and imaginary half-period $\omega_3$ satisfying $\operatorname{Im}\omega_3>0$, and for its hyperbolic degenerations (the $\coth$-type limit of the elliptic potential, as well as the hyperbolic near-neighbors model on the line with $\varphi(x)=\alpha\beta\coth(\beta x)$, $\beta>0$, $\omega>0$), determine the exact solution content of these models beyond the Jastrow-like ground state, bringing them to the same level of completeness as their rational and trigonometric predecessors. Concretely:

1. (Excited spectrum.) Determine whether eigenfunctions other than the Jastrow-like ground state can be computed in closed form, and compute them: either as explicit formulas, or through a quasi-exactly solvable structure, i.e. an explicit flag of finite-dimensional subspaces preserved by the Hamiltonian that can be diagonalized, in the sense in which this is known for the rational and trigonometric models.

2. (Spin counterparts.) Construct the spin (supersymmetric) generalizations of the elliptic and hyperbolic near-neighbors models, in which each particle carries an internal spin and the near-neighbors dynamics is coupled to spin-exchange operators, and derive from them the associated short-range spin chains by the freezing trick.

3. (Correlation functions.) Compute the correlation functions of the ground-state probability densities, in particular the $n$-point correlation functions and the distribution of spacings of consecutive levels — in the sense of Bogomolny, Gerland and Schmit, the particle coordinates of the ground-state density playing the role of energy levels, with the separations restricted to the ordered sector $s_i>0$ — for the elliptic density $\exp\big(-\tfrac{2\alpha\eta_1}{l}\sum_i s_i^2\big)\prod_i|\sigma(s_i)|^{2\alpha}$ with $s_i=x_i-x_{i+1}$ and $\sum_i s_i=l$, including their degeneration limits.

These three components jointly constitute the exact solution theory of the newly classified elliptic and hyperbolic families (spectrum, supersymmetric extension, and the statistical physics of the ground state), in the same sense in which the corresponding results are known for the rational and trigonometric near-neighbors models. The source paper's further proposal, namely extending the classification itself to Jastrow-like ground states of fixed range $r>1$ and to factorizations over other root systems such as $BC_N$, is a distinct companion problem (it generalizes the classification theorem rather than solving the classified models) and is not part of the completion boundary here.

References

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